Iterated Medial Triangle Subdivision in Surfaces of Constant Curvature

Abstract

Consider a geodesic triangle on a surface of constant curvature and subdivide it recursively into 4 triangles by joining the midpoints of its edges. We show the existence of a uniform $\delta>0$ such that, at any step of the subdivision, all the triangle angles lie in the interval $(\delta, \pi -\delta)$. Additionally, we exhibit stabilising behaviours for both angles and lengths as this subdivision progresses.